Definition of ≅ ∆`s Definition of Midpoint Isosceles ∆ Vertex Thms
... DATE __________ Per.___________ S.s.A. THEOREM (for Right ∆’s): If one angle (∠LHS) and two consecutive sides (LS & SH) of one triangle (∆LHS) are congruent to one angle (∠TWO) and two consecutive sides (TO & OW) of another (∆TWO), and the larger of the two congruent sides (LS > SH) are opposite the ...
... DATE __________ Per.___________ S.s.A. THEOREM (for Right ∆’s): If one angle (∠LHS) and two consecutive sides (LS & SH) of one triangle (∆LHS) are congruent to one angle (∠TWO) and two consecutive sides (TO & OW) of another (∆TWO), and the larger of the two congruent sides (LS > SH) are opposite the ...
9.15 Group Structures on Homotopy Classes of Maps
... Proof: Intuitively, think of S n+1 as the one point compactification of Rn+1 and notice that after removal of the point at which the identifications have been made, SS n opens up to become an open (n + 1)-disk. For a formal proof, write S k as I k /∂(I k ) and notice that both SS n and S n+1 becomes ...
... Proof: Intuitively, think of S n+1 as the one point compactification of Rn+1 and notice that after removal of the point at which the identifications have been made, SS n opens up to become an open (n + 1)-disk. For a formal proof, write S k as I k /∂(I k ) and notice that both SS n and S n+1 becomes ...
Dimension theory
... By Proposition 11.14, the polynomial g(n) has degree d(M � ) and χM m (n) has degree d(M �� ). Since both of these polynomials have positive leading terms, the leading terms cannot cancel and their sum has degree equal to the maximum of the two degrees. In the special case when M ∼ = M � , we use th ...
... By Proposition 11.14, the polynomial g(n) has degree d(M � ) and χM m (n) has degree d(M �� ). Since both of these polynomials have positive leading terms, the leading terms cannot cancel and their sum has degree equal to the maximum of the two degrees. In the special case when M ∼ = M � , we use th ...
Modules I: Basic definitions and constructions
... • If x = a + bi + cj + dk is a quaternion, its quaternionic conjugate is defined by x = a − bi − cj − dk. Quaternionic conjugation is an anti-automorphism of H, as can be checked by direct computation. • If R is a commutative ring, then the identity is both an automorphism and an antiautomorphism. ...
... • If x = a + bi + cj + dk is a quaternion, its quaternionic conjugate is defined by x = a − bi − cj − dk. Quaternionic conjugation is an anti-automorphism of H, as can be checked by direct computation. • If R is a commutative ring, then the identity is both an automorphism and an antiautomorphism. ...
Ulrich bundles on abelian surfaces
... Remarks. 1) There is no Ulrich line bundle on a general abelian surface X. Indeed, a line bundle M on X with χ(M ) = 0 satisfies c1 (M )2 = 0 by Riemann-Roch; since X is general, we have NS(X) = Z, hence M is algebraically equivalent to OX . Thus if L is an Ulrich line bundle, L(−1) and L(−2) must be ...
... Remarks. 1) There is no Ulrich line bundle on a general abelian surface X. Indeed, a line bundle M on X with χ(M ) = 0 satisfies c1 (M )2 = 0 by Riemann-Roch; since X is general, we have NS(X) = Z, hence M is algebraically equivalent to OX . Thus if L is an Ulrich line bundle, L(−1) and L(−2) must be ...
on end0m0rpb3sms of abelian topological groups
... (ii) X generates F(X) algebraically, and (hi) for every continuous map tpoiX into any topological group G such that
... (ii) X generates F(X) algebraically, and (hi) for every continuous map tpoiX into any topological group G such that
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... arbitrary quasi-separated scheme (or algebraic space) over k on which G acts (quasi-separated means that the diagonal embedding X ,→ X ×Spec k X is quasicompact; this is automatically satisfied when X is of finite type over k ). Assume further that the G-action on X is free, i.e., that the stabilize ...
... arbitrary quasi-separated scheme (or algebraic space) over k on which G acts (quasi-separated means that the diagonal embedding X ,→ X ×Spec k X is quasicompact; this is automatically satisfied when X is of finite type over k ). Assume further that the G-action on X is free, i.e., that the stabilize ...
1. Quick intro 2. Classifying spaces
... Let G be a topological group (i.e. a group which is also a topological space and whose operations are continuous maps) and let X be a G-space (i.e. a topological space equipped with a continuous G-action). We want to study the cohomology of X , keeping track of the G-action. A rst guess would be of ...
... Let G be a topological group (i.e. a group which is also a topological space and whose operations are continuous maps) and let X be a G-space (i.e. a topological space equipped with a continuous G-action). We want to study the cohomology of X , keeping track of the G-action. A rst guess would be of ...
Homework sheet 6
... Please complete all the questions. 1. Let X → Y be a morphism of affine varieties over a field k. Show that the induced morphism k[Y ] → k[X] on rings of regular functions is injective if and only if the original morphism X → Y has dense image. 2. Prove that the Segre embedding from Pm (Ω) → Pn (Ω) ...
... Please complete all the questions. 1. Let X → Y be a morphism of affine varieties over a field k. Show that the induced morphism k[Y ] → k[X] on rings of regular functions is injective if and only if the original morphism X → Y has dense image. 2. Prove that the Segre embedding from Pm (Ω) → Pn (Ω) ...
Categories and functors
... as an immediate consequence of basic definitions that all left adjoint functors are right exact and similarly for right adjoint functors. Definition 22.1. A category C is a collection Ob(C) of objects A and morphism sets HomC (A, B) whose elements are written f : A → B (with A = s(f ), B = t(f )) to ...
... as an immediate consequence of basic definitions that all left adjoint functors are right exact and similarly for right adjoint functors. Definition 22.1. A category C is a collection Ob(C) of objects A and morphism sets HomC (A, B) whose elements are written f : A → B (with A = s(f ), B = t(f )) to ...
PERIODS OF GENERIC TORSORS OF GROUPS OF
... can embed Q into GLn for some positive integer n, resolutions of Q exist. The generic fiber of the morphism φ : P → S is a Q-torsor over the function field F (S) of S, moreover it is a generic Q-torsor (see [1, section 6] and [5]). In this paper by a generic Q-torsor, we mean a generic Q-torsor that ...
... can embed Q into GLn for some positive integer n, resolutions of Q exist. The generic fiber of the morphism φ : P → S is a Q-torsor over the function field F (S) of S, moreover it is a generic Q-torsor (see [1, section 6] and [5]). In this paper by a generic Q-torsor, we mean a generic Q-torsor that ...
Solutions Sheet 3
... nor an epimorphism, as for example for the zero morphism Z → Z in the category of Z-modules, we get a functor that preserves neither monomorphisms nor epimorphisms. (Explanation: The property of being a mono- or epimorphism involves a quantification over all objects of a category. So if a functor is ...
... nor an epimorphism, as for example for the zero morphism Z → Z in the category of Z-modules, we get a functor that preserves neither monomorphisms nor epimorphisms. (Explanation: The property of being a mono- or epimorphism involves a quantification over all objects of a category. So if a functor is ...
ORTHOGONAL BUNDLES OVER CURVES IN CHARACTERISTIC
... generally for SO(n) with n ≥ 7 after adding direct summands of hyperbolic planes. Note that Behrend’s conjecture holds for SO(n) with n ≤ 6 because of the exceptional isomorphisms with other classical groups. The first three sections are quite elementary and recall well-known facts on quadratic form ...
... generally for SO(n) with n ≥ 7 after adding direct summands of hyperbolic planes. Note that Behrend’s conjecture holds for SO(n) with n ≤ 6 because of the exceptional isomorphisms with other classical groups. The first three sections are quite elementary and recall well-known facts on quadratic form ...
3. Modules
... (a) Let V be a vector space over a field K. If V has finite dimension n, there is a chain 0 = V0 ( V1 ( · · · ( Vn = V of subspaces of V with dimK Vi = i for all i. Obviously, this chain cannot be refined. Hence it is a composition series for V , and we conclude by Exercise 3.19 (b) that l(V ) = n = ...
... (a) Let V be a vector space over a field K. If V has finite dimension n, there is a chain 0 = V0 ( V1 ( · · · ( Vn = V of subspaces of V with dimK Vi = i for all i. Obviously, this chain cannot be refined. Hence it is a composition series for V , and we conclude by Exercise 3.19 (b) that l(V ) = n = ...
ON THE POPOV-POMMERENING CONJECTURE FOR GROUPS
... and (hence) all its conjugates are Grosshans subgroups of GL5(zc). Other subgroups of GL5(k) (n < 5) normalized by a maximal torus can be checked in the same spirit. (In fact, they can be proved to be Grosshans using various known results, cf. [G3], [Po2], [T], etc. [Po2] gives a more detailed accou ...
... and (hence) all its conjugates are Grosshans subgroups of GL5(zc). Other subgroups of GL5(k) (n < 5) normalized by a maximal torus can be checked in the same spirit. (In fact, they can be proved to be Grosshans using various known results, cf. [G3], [Po2], [T], etc. [Po2] gives a more detailed accou ...
N AS AN AEC Very Preliminary We show the concept of an Abstract
... (2) Can the question of whether a class(e.g. Whitehead groups) is a P CΓclass (defined as the reducts of models of say a countable theory that omitting a family of types be independent of ZFC? (Note that under V = L, ‘Whitehead=free’ and the class is easily P CΓ. Lemma 1.16. For a cotorsion Abelian ...
... (2) Can the question of whether a class(e.g. Whitehead groups) is a P CΓclass (defined as the reducts of models of say a countable theory that omitting a family of types be independent of ZFC? (Note that under V = L, ‘Whitehead=free’ and the class is easily P CΓ. Lemma 1.16. For a cotorsion Abelian ...
Spencer Bloch: The proof of the Mordell Conjecture
... one fixes a suitable origin P0, the procedure for multiplying a point R by 2 is given in fig. 2. Assuming the original point was not of finite order for the group law, iteration will yield an infinite set of solutions. One consequence of Faltings' work is that for F non-singular of degree/> 4, no su ...
... one fixes a suitable origin P0, the procedure for multiplying a point R by 2 is given in fig. 2. Assuming the original point was not of finite order for the group law, iteration will yield an infinite set of solutions. One consequence of Faltings' work is that for F non-singular of degree/> 4, no su ...
Math 210B. Absolute Galois groups and fundamental groups 1
... 1.1. Review of covering spaces. Let X be a path-connected topological space, and x0 ∈ X a point. The fundamental group π1 (X, x0 ) classifying homotopy classes of loops in X based at x0 is a key tool in the study of covering spaces q : X 0 → X, at least when X is “nice” (which we shall now assume; t ...
... 1.1. Review of covering spaces. Let X be a path-connected topological space, and x0 ∈ X a point. The fundamental group π1 (X, x0 ) classifying homotopy classes of loops in X based at x0 is a key tool in the study of covering spaces q : X 0 → X, at least when X is “nice” (which we shall now assume; t ...
The Picard group
... ? X(Q) X(Fp ) o X(Qp ) o Figure 1: Some of the sets of points associated to a Diophantine equation • On X(C), we have all the tools available to study complex analytic varieties. For example, X(C) has cohomology groups which give much information about its geometry, and these come with Hodge decompo ...
... ? X(Q) X(Fp ) o X(Qp ) o Figure 1: Some of the sets of points associated to a Diophantine equation • On X(C), we have all the tools available to study complex analytic varieties. For example, X(C) has cohomology groups which give much information about its geometry, and these come with Hodge decompo ...
Algebraic Transformation Groups and Algebraic Varieties
... Let G be a connected linear algebraic group over C and let H a closed algebraic subgroup. A fundamental problem in the study of homogeneous spaces is to describe, characterize, or classify those quotients G/H that are affine varieties. While cohomological characterizations of affine G/H are possible, th ...
... Let G be a connected linear algebraic group over C and let H a closed algebraic subgroup. A fundamental problem in the study of homogeneous spaces is to describe, characterize, or classify those quotients G/H that are affine varieties. While cohomological characterizations of affine G/H are possible, th ...
THE BRAUER GROUP: A SURVEY Introduction Notation
... algebras over R. We denote by Br(R) := (AzR / ∼Br ) the Brauer group of R. 3. Brauer Group of a Scheme Let X be a scheme (over a field F , or over a scheme S). In a usual way, we extend a notion for commutative rings to schemes. Definition 3.1. An Azumaya algebra A over X is a sheaf of OX -algebras ...
... algebras over R. We denote by Br(R) := (AzR / ∼Br ) the Brauer group of R. 3. Brauer Group of a Scheme Let X be a scheme (over a field F , or over a scheme S). In a usual way, we extend a notion for commutative rings to schemes. Definition 3.1. An Azumaya algebra A over X is a sheaf of OX -algebras ...
Notes 1
... Note A field can be defined to satisfy (i) X ∈ F, (ii) If A ∈ F then Ac ∈ F, (iii) If A, B ∈ F then A ∪ B ∈ F . Similarly for σ-fields. *Verification that (i) X ∈ F , (ii) A, B ∈ R ⇒ A ∪ B ∈ R, and (iii) A, B ∈ R ⇒ A \ B ∈ R are equivalent to (i’) X ∈ F , (ii’) A, B ∈ R ⇒ A ∪ B ∈ R,and (iii’) if A ∈ ...
... Note A field can be defined to satisfy (i) X ∈ F, (ii) If A ∈ F then Ac ∈ F, (iii) If A, B ∈ F then A ∪ B ∈ F . Similarly for σ-fields. *Verification that (i) X ∈ F , (ii) A, B ∈ R ⇒ A ∪ B ∈ R, and (iii) A, B ∈ R ⇒ A \ B ∈ R are equivalent to (i’) X ∈ F , (ii’) A, B ∈ R ⇒ A ∪ B ∈ R,and (iii’) if A ∈ ...
Algebraic numbers and algebraic integers
... Now, suppose that the algebraic numbers α and β respectively satisfy the polynomials f (T ) and g(U ) over Q. Then the condition R(f (T ), R(g(U ), T + U − V )) = 0 first eliminates U from the simultaneous conditions g(U ) = 0, T = U + V , leaving a polynomial condition h(T, V ) = 0, and then it eli ...
... Now, suppose that the algebraic numbers α and β respectively satisfy the polynomials f (T ) and g(U ) over Q. Then the condition R(f (T ), R(g(U ), T + U − V )) = 0 first eliminates U from the simultaneous conditions g(U ) = 0, T = U + V , leaving a polynomial condition h(T, V ) = 0, and then it eli ...
The braid group action on the set of exceptional sequences
... We consider exceptional sequences of objects in a triangulated category D b , as introduced in the first lecture. In particular, we assume that all left and right mutations are defined in this category. We denote by Hom • (E, F ) the finite dimensional vector space ⊕l Hom (E, F [l]). Moreover, we wr ...
... We consider exceptional sequences of objects in a triangulated category D b , as introduced in the first lecture. In particular, we assume that all left and right mutations are defined in this category. We denote by Hom • (E, F ) the finite dimensional vector space ⊕l Hom (E, F [l]). Moreover, we wr ...
Algebraic Geometry 3-Homework 11 1. a. Let O be a noetherian
... is a line in P2 considered as a Cartier divisor. Show that deg(α) is welldefined (i.e., independent of the choice of `). b. Since P2 is smooth over k, we have Div(P2 ) = Z1 (P2 ), so for each pair of 1-cycles, α, β ∈ Z1 (P2 ), the intersection produce α · β ∈ CH0 (P2 ) is defined by considering α as ...
... is a line in P2 considered as a Cartier divisor. Show that deg(α) is welldefined (i.e., independent of the choice of `). b. Since P2 is smooth over k, we have Div(P2 ) = Z1 (P2 ), so for each pair of 1-cycles, α, β ∈ Z1 (P2 ), the intersection produce α · β ∈ CH0 (P2 ) is defined by considering α as ...